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Chapter 8: Problem 366

In the following exercises, solve. $$ \frac{n}{91}=\frac{8}{13} $$

Short Answer

Expert verified
n = 56

Step by step solution

01

- Set up the equation

Start with the given equation: \[ \frac{n}{91} = \frac{8}{13} \]
02

- Cross-multiply

To solve for \( n \), cross-multiply the fractions. This means multiplying the numerator of one fraction by the denominator of the other: \[ n \times 13 = 8 \times 91 \]
03

- Simplify the products

Calculate the products: \[ n \times 13 = 728 \]
04

- Isolate \( n \)

To solve for \( n \), divide both sides of the equation by 13: \[ n = \frac{728}{13} \]
05

- Simplify the fraction

Divide 728 by 13 to find \( n \): \[ n = 56 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cross-multiplication
To solve proportions, cross-multiplication is a powerful technique. Start with the equation you need to solve, like \(\frac{n}{91} = \frac{8}{13}\). Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other. For our example, multiply \(n\) by 13 and 8 by 91. This operation eliminates the fractions and creates a simpler equation: \(n \times 13 = 8 \times 91\).

Remember, cross-multiplication is useful when both sides of the equation are fractions. This method helps you quickly get rid of the fractions, making it easier to isolate the variable in the next steps.
isolate variable
After cross-multiplying, the next step is to isolate the variable. Here our equation is \(n \times 13 = 728\). To isolate \(n\), divide both sides by 13. Isolating the variable means getting the variable on its own. It's an essential step in solving equations.

For our example: \[ n = \frac{728}{13} = 56. \]

Always perform the same operation on both sides of the equation. This keeps the equation balanced and leads to the correct value for the variable.
simplifying fractions
Sometimes, our solutions involve fractions, and simplifying them is crucial. Simplifying means expressing a fraction in its simplest form. Divide the numerator and the denominator by their greatest common divisor (GCD).

In our example, we ended up with \(n = \frac{728}{13}\). We found that dividing 728 by 13 gives us a whole number, which is 56.

While our fraction wasn鈥檛 complicated to simplify, always check if further simplification is possible. This practice ensures your final answer is as straightforward as possible.

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